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A set of master variables for the two-star random graph

Pawat Akara-pipattana, Oleg Evnin

TL;DR

The article develops a master-variable framework for the two-star random graph, introducing auxiliary variables $x_J$ that couple to exponential sums of the Hubbard-Stratonovich fields. In the dense regime ($\beta\sim 1/N$), the method recovers the Park–Newman mean-field solution while explicitly controlling $1/N$ corrections to the free energy. In the sparse regime ($\alpha\sim \tfrac{1}{2}\log N$), it yields a compact derivation of the Annibale–Courtney solution via a one-dimensional saddle in $x_1$, with the master-variable condensation $X$ linked to the degree distribution through $X= c e^{-\beta}\langle e^{-2\beta k}\rangle$ and related thermodynamic identities. The results provide a transparent, non-functional-integral route to known solutions and offer a principled way to quantify finite-size effects, with Monte Carlo validation and consistency checks such as particle-hole duality. The master-variable perspective is argued to generalize to other exponential random-graph models and large-N problems.

Abstract

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such `master variables' are usually highly desirable in treatments of `large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.

A set of master variables for the two-star random graph

TL;DR

The article develops a master-variable framework for the two-star random graph, introducing auxiliary variables that couple to exponential sums of the Hubbard-Stratonovich fields. In the dense regime (), the method recovers the Park–Newman mean-field solution while explicitly controlling corrections to the free energy. In the sparse regime (), it yields a compact derivation of the Annibale–Courtney solution via a one-dimensional saddle in , with the master-variable condensation linked to the degree distribution through and related thermodynamic identities. The results provide a transparent, non-functional-integral route to known solutions and offer a principled way to quantify finite-size effects, with Monte Carlo validation and consistency checks such as particle-hole duality. The master-variable perspective is argued to generalize to other exponential random-graph models and large-N problems.

Abstract

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such `master variables' are usually highly desirable in treatments of `large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.

Paper Structure

This paper contains 11 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The two-star random graph and its auxiliary field representation
  3. The dense regime at leading order
  4. The dense regime: subleading corrections
  5. The sparse regime
  6. The sparse regime: condensation of the master variable
  7. Conclusions
  8. Naive expansion around the Park-Newman solution
  9. Some useful formulas
  10. Particle-hole duality
  11. Monte Carlo sampling

Figures (1)

  • Figure 1: The difference $\delta V$ between degree variance and the leading order prediction $V_0$ given by (\ref{['defV0']}) as a function of $\alpha$. The crosses represent numerical measurements from Monte Carlo sampling for graphs with $N=200$ vertices, the dots are the same for $N=50$ vertices, deviating from the analytic curve slightly stronger because of higher $1/N$ corrections, while the solid lines are the analytic prediction (\ref{['defVcorr']}). Two different values are presented: $B=1.3$ on the left and $B=-2$ on the right.